SAN FERNANDO, Calif.—The Oscar-nominated designers at Legacy Effects have outfitted such memorable movie warriors as The Terminator, RoboCop, Captain America and Iron Man.

The special-effects company is now at work on what seems a mission impossible: Building an Iron Man-style suit to protect and propel elite U.S. troops by encasing them in body armor equipped with an agile exoskeleton to enable troops to carry hundreds of pounds of...